Sequential move games are played by two or more players for two or more periods

◮

Ultimatum game is a sequential game

◮

Chess is a sequential game

◮

Paper-Rock-Scissors is not a sequential game

2 / 41

Representing Sequential Games: Game Trees

◮

Sequential move games are normally presented as game trees.

◮

Game trees are often referred to as extensive form games.

◮

Think of these as a possible paths of players’ actions and outcomes ◮

Game trees reveal the players, their actions, the timing of their actions and the payoffs.

◮

Game trees vs. decision trees: Game trees are joint decision trees for all the players in a game.

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Sir Richard Branson

4 / 41

Game Trees
Example

Consider the phone company choice game represented by the following game tree with two players, Husband (H) and Wife (W):
H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

So if the actions are H:{Telstra} and W:{Virgin}, H gets 3 and W gets 5. That is, wife prefers Virgin if husband prefers Telstra.

5 / 41

Game Trees
Game characteristics

◮

This is a non-zero sum game.
If actions are H:{Telstra} and W:{Virgin}, then H gets 3 and
W gets 5
If actions are H:{Virgin} and W:{Virgin}, then H gets 6 and
W gets 6
So we see pay-offs from actions do not result in one player’s gain = other player’s loss. Here, both do better if H:{Virgin} and W:{Virgin} then if H:{Telstra} and W:{Virgin}!

◮

This is a complete information game.
W sees all possible previous moves by all other players (H)
H can anticipate any reaction of W to H’s choice

6 / 41

Game Trees
Characterization of a Game Tree

◮

Players: Two players – Husband (H) and Wife (W)

◮

Nodes: Three decision nodes and four terminal nodes

◮

Timing: Sequential, player H moves first, then player W

◮

Outcomes and Payoffs: At each terminal node, payoffs for all players are listed for that sequence of moves. They are normally listed in the order of who moves first.

◮

Actions are moves taken at decision nodes, where each branch represents a possible action.

◮

Strategies: action plans that describe a player’s actions at all of his/her decision nodes

7 / 41

2 Things that Drive Your Lecturer Nuts
STRATEGIES ARE ACTION PLANS!

H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

◮

Example strategy: H Virgin) and W is (Virgin, Telstra)

◮

They are NOT payoffs (i.e, (4,3) from the example strategy)

◮

They are NOT the sequence of decisions implied by action plans (i.e., (Virgin, Telstra) from the example strategy)

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Game Trees
Backward Induction

◮

A subgame is a game comprising a portion of a larger game, starting at a non-initial node of the larger game.

◮

Backward induction asks to start at the final subgames, and to work backwards towards the initial node.
(That’s why backward induction is also known as rollback.)

◮

Rational player selects in every subgame the move that maximises their own payoff.

9 / 41

Game Trees
Example

The phone company choice game has three subgames:
◮

The two subgames for each of the possible (W) choices

◮

The one subgame for the one (H) choice (i.e., the entire game!) H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

10 / 41

Game Trees
Backward Induction
◮

To start solving the phone company game, consider the two final subgames:
When player H has chosen Telstra, player W gets payoff 4 if she chooses Telstra and 5 if she chooses Virgin.
When Player H has chosen Virgin, player W gets payoff 3 if she chooses Telstra and 6 if she chooses Virgin.

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